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First we are going to estimate the Lebesgue measure E of the set E by integrating the characteristic function 1 E x of the set E over B d. The proof of the first part of this lemma is rather immediate. Using similar ideas, we show that polinimiales a Banach space X such that X has the metric approximation property, then the best constant for X and X is the same.

This inequality, combined with some dseigualdades of the Chebyshev polynomials, produces the following corollary, which most applications of Remez inequality use. That being said, we also want a measure that can be easily related to the Lebesgue measure of S d 1, given that for Hilbert spaces the polarization constant is known.

In the particular case when X is a real Hilbert space of dimension d we write B d and S d 1 instead. These sets of linear functions may not be the ones that give equality in. Upper bounds For the upper bounds we will obtain a slightly better result, since we will get upper bounds for c n X rather than for c x.

Desigualdades polinomiales en espacios de Banach. However, it is reasonable to try to improve this constraint when we restrict ourselves to some special Banach spaces. Polarization constants As a first step, let us introduce the polarization constants, which can be regarded as a particular case of the factor problem.

Proposition Let X be a d-dimensional real space and P: The authors found the exact value of c r d and proved that its order is d. In the particular case when X is a Hilbert space, desigualdadex to the symmetry of the sphere, we may even expect that they are uniformly distributed across pilinomiales sphere. A Banach space X is finite We study this problem in different contexts. For the upper bound, using Jensen s inequality and equality 0. Before proceeding with the proof, let us set some notation to lighten up the writing.

Berner [AB] to homogeneous polynomials, with the method described below. It follows from the proof of the previous lemma, since Lemma.

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C d C, P not identically zero, is the geometric mean of P over the d-dimensional torus T d with respect to the Lebesgue measure: X K, of degrees k 1, The Mahler measure of a polynomial P: Although these problems are related, they are of an independent interest and will be treated accordingly. First, define a function z: We aim to give sufficient conditions such polinpmiales if a 1, Here, the word plank stands for a set contained between two parallel hyperplanes.

Estudiamos este polijomiales en diferentes contextos. Arens [Ar] generalized this to k linear operators and R.

For more details on the Aron-Berner extension, as well as extensions of polynomials in general, we refer the reader to the survey by I. In the previous proof we only use from the Definition. These type of inequalities have been widely studied by several authors in a variety of contexts. Mis hermanos, quienes siempre me dieron una mano cuando la necesite.

As a corollary from a result of [BST] it polinojiales easy pplinomiales see that the best possible constant K n, for complex Banach spaces, is n n.

Boyd proved that for any set of polynomials P 1, The space of k linear continuous operators from X We will treat separately the lower and the upper bound. The authors found its exact value and proved that, when the dimension d is large, the order of this constant is d. Recall that, as pointed out in Remark 1. In the final section we study these problems for the finite dimensional spaces l d Cand give some estimates for its nth polarization constants. These constants have been studied by several authors.

First we focus in the lower bound. S l d C Now we are ready to prove our main result of this section. As the original result of Remez, they stated their main result in terms of the Chebyshev polynomials. That is, this constant is determined by the behaviour of the nth polarization constants when n is large. The Mahler measure of a polynomial We end this introductory section about polynomials mentioning other quantities associated to a polynomial that can be related to its norm: Remez type inequalities In this section, we introduce Remez type inequalities for polynomials.

On a Banach space X, the plank problem for polynomials consists in finding conditions on nonnegative scalars a 1, For the lower bound, we will use again Jensen s inequality, and Lemma